sign change conjugates Dirichlet eta function

s = N[ZetaZero[1], 100] - 1/2 + 1/3;

sign = -1;
Sum[(-1)^k*(sign*Exp[-s/(Im[s]/Pi)])^(Log[k]*(Im[s]/Pi)), {k, 1,
  Infinity}]

-Conjugate[Zeta[Re[s]]*(1 - 1/2^(Re[s] - 1))]

sign = 1;
Sum[(-1)^k*(sign*Exp[-s/(Im[s]/Pi)])^(Log[k]*(Im[s]/Pi)), {k, 1,
  Infinity}]

-Conjugate[Zeta[s]*(1 - 1/2^(s - 1))]